Olga Taussky-Todd

How to use this course

Olga Taussky-Todd wrote about 300 papers across algebraic number theory, group theory, numerical analysis and, above all, matrix theory. She began in the most abstract corner of 1920s mathematics, class field theory, and was pulled by war work into the practical problem of computing eigenvalues. She then spent four decades showing that the two worlds belong together: integer matrices encode ideal classes, and questions about commuting matrices are questions about algebra. She liked to call herself a torchbearer for matrix theory.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 and 05 follow her number-theoretic beginnings (1930–1937). Lessons 06 to 08 come from the matrix theory she took up during and after the war. Lessons 09 and 10 show the bridges she built back to number theory.

Timeline

Olmütz, Vienna, Linz, 1906–1925

Olga Taussky was born on 30 August 1906 in Olmütz, a Moravian town in the Austro-Hungarian Empire, into a Jewish family. Her father, Julius David Taussky, was an industrial chemist who also wrote for newspapers. Her mother, Ida Pollach, had been raised to run a household. Olga was the middle of three daughters, born three years apart. Her elder sister Ilona became a chemist; her younger sister Hertha became a pharmacist and later a clinical chemist at Cornell University Medical College.

When Olga was three the family moved to Vienna, where she went to primary school. Life there grew very hard during the First World War, and in 1916 the family moved to Linz in Upper Austria, where her father became director of a vinegar factory. At school she first loved Latin and writing, and only gradually turned to mathematics.

Her father died during her final school year. From about fifteen she had tutored other pupils, and the income now helped the family. Her father had wanted her to take over his chemical work, but her elder sister did so instead, and Olga was free to choose. In the autumn of 1925 she entered the University of Vienna intending to study chemistry, and soon made mathematics her main subject.

Vienna and Göttingen, 1925–1934

Vienna in the late 1920s had a brilliant mathematics department. Taussky studied with Philipp Furtwängler, Hans Hahn, Wilhelm Wirtinger, Karl Menger and Eduard Helly. Kurt Gödel was a fellow student, and she attended meetings of the Vienna Circle, one of the first women to do so.

Furtwängler, a number theorist, became her advisor. He was then finishing the proof of the principal ideal theorem, a conjecture of David Hilbert that Emil Artin had reduced to a statement about finite groups in 1929. Taussky’s thesis, Über eine Verschärfung des Hauptidealsatzes (On a sharpening of the principal ideal theorem), studied the finer question of which ideal classes become principal in intermediate fields. She completed the work partly in Zürich and passed her oral examination in Vienna in 1930. The thesis appeared in Crelle’s Journal in 1932. Lesson 04 explains the mathematics.

Göttingen

On Hahn’s recommendation, Richard Courant hired her in 1931 as an assistant at Göttingen. Her main task was to help edit the first volume of Hilbert’s collected works, on number theory, with Wilhelm Magnus and Helmut Ulm. The volume was to be presented to Hilbert for his seventieth birthday in 1932. The editors found and corrected many errors in Hilbert’s papers. She also edited Artin’s lectures on class field theory, helped Courant with a course on differential equations, and took part in Emmy Noether’s seminar.

In the summer of 1932 Courant advised her not to return, because of the political situation. She went back to Vienna as an assistant to Hahn and Menger. In this period she began working with Arnold Scholz, a student of Issai Schur, on class field towers. Their joint paper on principal ideals of cubic class fields of imaginary quadratic fields appeared in 1934 (Lesson 05).

Bryn Mawr, Cambridge, London and the war, 1934–1946

In 1934, with help from Oswald Veblen, Taussky received a fellowship to Bryn Mawr College in Pennsylvania. Emmy Noether, expelled from Göttingen, was there too, and Taussky spent the academic year 1934–35 working near her. Noether died in April 1935. That autumn Taussky took up a research fellowship at Girton College, Cambridge.

In 1937, with help from G. H. Hardy, she obtained a teaching post at Westfield College in the University of London. There she met John Todd, known as Jack, an Irish mathematician from County Down who was teaching at another London college. They married on 29 September 1938. Their careers would be intertwined from then on; Todd became one of the founders of modern numerical analysis.

War work

During the war Westfield was evacuated to Oxford, and Jack’s family home was in Belfast; the couple moved between the two. In 1942–43 she supervised the doctoral work of Hanna Neumann on combinatorial group theory. From 1943 to 1946 she worked at the National Physical Laboratory in Teddington for the Ministry of Aircraft Production, on flutter: the self-excited vibration of aircraft wings that can destroy a plane in seconds. Flutter problems lead to large systems of linear differential equations, and the question is whether the eigenvalues of certain matrices have negative real parts. Taussky found herself computing and bounding eigenvalues, and she developed ways to shrink Gershgorin discs by similarity transformations to cut down the arithmetic. Her wartime reports appeared under the name Olga Todd.

The experience changed her mathematics. She saw that matrix theory, then regarded by many pure mathematicians as a minor technical subject, was full of deep questions and connected naturally to her algebra. Lessons 06 to 08 come from this turn.

The principal ideal theorem and capitulation

For the general reader. In some number systems, numbers do not factor uniquely into primes, and the class group measures the failure. Hilbert conjectured that every such system sits inside a larger one, its Hilbert class field, in which every ideal of the smaller system becomes principal. Artin turned this into a statement about finite groups and Furtwängler proved it in 1930. Taussky’s thesis and later work asked the next question: which ideals become principal in which intermediate fields?

Class fields

Let \(K\) be a number field with ideal class group \(\mathrm{Cl}(K)\). The Hilbert class field \(H\) is the maximal abelian extension of \(K\) unramified at all places. Class field theory gives an isomorphism \(\mathrm{Cl}(K)\cong\mathrm{Gal}(H/K)\) (the Artin map), under which a prime ideal \(\mathfrak p\) maps to its Frobenius element. For an extension \(L/K\) there is a map \(j_{L/K}:\mathrm{Cl}(K)\to\mathrm{Cl}(L)\), \([\mathfrak a]\mapsto[\mathfrak a\mathcal O_L]\). Classes in its kernel are said to capitulate in \(L\).

Artin’s reduction

Let \(H_2\) be the Hilbert class field of \(H\), and \(G=\mathrm{Gal}(H_2/K)\). Then \(G'=\mathrm{Gal}(H_2/H)\), so \(G/G'\cong\mathrm{Cl}(K)\) and \(G'/G''=G'\cong\mathrm{Cl}(H)\). Artin showed that under these isomorphisms \(j_{H/K}\) becomes the transfer \(V:G/G'\to G'\). For a subgroup \(U\) of finite index in \(G\), with right coset representatives \(t_1,\dots,t_n\), the transfer is \(V(g)=\prod_i t_i\,g\,t_{\sigma(i)}^{-1}\) modulo \(U'\), where \(t_ig\in Ut_{\sigma(i)}\).

Shokichi Iyanaga, Ernst Witt and Hans Zassenhaus later gave simpler proofs.

Partial capitulation

For an unramified cyclic extension \(L/K\) of prime degree \(p\), Hilbert’s Theorem 94 says that \(\ker j_{L/K}\) is non-trivial. By class field theory, \(L\) corresponds to a subgroup \(N_{L/K}\mathrm{Cl}(L)\) of index \(p\) in \(\mathrm{Cl}(K)\). In a 1970 paper Taussky separated two cases:

The distinction controls how the capitulation kernels of the \(p+1\) cyclic unramified extensions (when the \(p\)-class group has rank 2) fit together, and it remains a basic tool in computing capitulation patterns and class field towers.

Class field towers and 2-groups

For the general reader. If a field has a non-trivial class group, pass to its Hilbert class field; that field has its own class group and its own Hilbert class field, and so on. Does the tower always stop at a field with unique factorization? Philipp Furtwängler asked this question. Taussky and Arnold Scholz computed the first non-trivial cases in 1934, and Taussky proved a group theorem showing that some towers stop after two steps. In 1964 Evgeny Golod and Igor Shafarevich showed that some towers never stop.

Fix a prime \(p\). The \(p\)-class field tower of \(K\) is \(K=K^{(0)}\subseteq K^{(1)}\subseteq K^{(2)}\subseteq\cdots\), where \(K^{(i+1)}\) is the maximal unramified abelian \(p\)-extension of \(K^{(i)}\). Let \(G=\mathrm{Gal}(K^{(\infty)}/K)\), the Galois group of the union. Then \(\mathrm{Gal}(K^{(i)}/K)\cong G/G^{(i)}\), where \(G^{(i)}\) is the \(i\)-th derived subgroup, and the tower has length \(\ell\) exactly when \(G\) is a finite \(p\)-group of derived length \(\ell\). In particular \(G/G'\) is the \(p\)-part of \(\mathrm{Cl}(K)\).

Proof sketch. If \(G/G'\) is cyclic, then \(G/\Phi(G)\) is cyclic, so \(G\) is cyclic by the Burnside basis theorem, and \(G'=1\), \(|G|=4\). Otherwise \(G\) is generated by two elements \(a,b\) and \(G'\) is the normal closure of \(c=[a,b]\). Induct on \(|G|\): if \(G'\neq1\), choose \(N\le G'\cap Z(G)\) of order 2. Then \(G/N\) satisfies the hypothesis, so by induction \((G/N)'=G'/N\) is cyclic. One then shows that \(G\) contains a cyclic subgroup of index 2 and reads off the result from the classification of 2-groups with a cyclic maximal subgroup.

Apply the theorem to \(G/G''=\mathrm{Gal}(K^{(2)}/K)\): its derived subgroup \(G'/G''\) is cyclic. Now apply it to \(G/G'''\): its derived subgroup \(G'/G'''\) is cyclic, hence abelian, so \(G''=G'''\), and since a non-trivial pro-\(p\) group always has a non-trivial abelian quotient, \(G''=1\). Scholz and Taussky studied the analogous question for \(p=3\), with 3-class group of type \((3,3)\), where the groups are far more varied. Their 1934 paper computed how ideal classes capitulate in the four unramified cubic extensions of such an imaginary quadratic field and used this to study the second stage of the tower.

By Gauss’s genus theory, the 2-rank of the class group of a quadratic field with discriminant \(d\) is \(t-1\), where \(t\) is the number of distinct primes dividing \(d\). The hypothesis therefore says the 2-class group has rank at least 5, and the proof is an inequality between the numbers of generators and relations of a finite \(p\)-group. Furtwängler’s question has a negative answer.

Gershgorin discs and irreducibility

For the general reader. Finding the eigenvalues of a large matrix exactly is hard, but often one only needs to know roughly where they are. In 1931 Semyon Gershgorin gave a simple answer: draw a disc around each diagonal entry, with radius the sum of the other entries in its row; the eigenvalues lie in those discs. Taussky made the theorem known, sharpened it, and used it constantly. Her 1949 paper on what she called a recurring theorem on determinants traced how often the same idea had been rediscovered.

Let \(A=(a_{ij})\in M_n(\mathbb C)\), \(R_i=\sum_{j\neq i}|a_{ij}|\), and \(D_i=\{z:|z-a_{ii}|\le R_i\}\).

Proof. Let \(Ax=\lambda x\), \(x\neq0\), and choose \(i\) with \(|x_i|=\max_j|x_j|\). Row \(i\) gives \((\lambda-a_{ii})x_i=\sum_{j\neq i}a_{ij}x_j\), so \(|\lambda-a_{ii}|\le\sum_{j\neq i}|a_{ij}|\,|x_j|/|x_i|\le R_i\). For the counting statement, let \(A(t)=D+t(A-D)\), \(D=\mathrm{diag}(a_{ii})\). The discs of \(A(t)\) have radii \(tR_i\), so the two unions stay disjoint for \(t\in[0,1]\); eigenvalues depend continuously on \(t\) and cannot cross the gap, and at \(t=0\) exactly \(k\) of them lie in the first union.

A matrix is irreducible if no permutation of rows and columns (the same on both) brings it to block upper triangular form; equivalently, its directed graph, with an edge \(i\to j\) when \(a_{ij}\neq0\), is strongly connected.

Proof sketch. In the proof above, \(\lambda\notin\operatorname{int}D_i\) for every \(i\), so all inequalities are equalities for every index \(i\) where \(|x_i|\) is maximal: \(|\lambda-a_{ii}|=R_i\), and \(|x_j|=|x_i|\) whenever \(a_{ij}\neq0\). By strong connectivity, the maximum spreads from \(i\) to all indices, so equality holds for every row. For the corollary, \(0\) would have to lie on every circle \(|z-a_{ii}|=R_i\), contradicting \(|a_{ii}|>R_i\) for some \(i\).

Shrinking discs

For a positive diagonal matrix \(P=\mathrm{diag}(p_1,\dots,p_n)\), \(P^{-1}AP\) has the same eigenvalues and entries \(a_{ij}p_j/p_i\). Choosing \(P\) well isolates an eigenvalue in a small disc. Taussky used such similarity transformations in her flutter computations to obtain good eigenvalue bounds with little arithmetic.

Stability and the Lyapunov equation

For the general reader. An aircraft wing in airflow obeys a system of linear differential equations. The motion dies away if every eigenvalue of the system matrix has negative real part, and grows catastrophically if one does not. This is the flutter problem Taussky worked on during the war. Aleksandr Lyapunov had shown in 1892 that stability can be tested by solving a linear matrix equation instead of computing eigenvalues. Taussky helped bring this theory into matrix analysis, where it grew into the theory of inertia.

A matrix \(A\in M_n(\mathbb C)\) is stable if every eigenvalue has negative real part. Then every solution of \(\dot x=Ax\), namely \(x(t)=e^{tA}x(0)\), tends to 0. A second-order system \(M\ddot q+D\dot q+Kq=0\), the form of flutter equations, becomes first order with \(x=(q,\dot q)\) and \(A=\begin{pmatrix}0&I\\-M^{-1}K&-M^{-1}D\end{pmatrix}\).

Proof of “if”. Let \(Ax=\lambda x\), \(x\neq0\). Then \(x^*(A^*P+PA)x=(\bar\lambda+\lambda)\,x^*Px=2\operatorname{Re}\lambda\;x^*Px\). The left side is negative and \(x^*Px>0\), so \(\operatorname{Re}\lambda<0\). “Only if”: if \(A\) is stable the integral converges, and differentiating \(e^{tA^*}Qe^{tA}\) and integrating from 0 to \(\infty\) gives \(A^*P+PA=-Q\); positivity is clear from the integrand.

The inertia of \(A\) is the triple \((\pi,\nu,\delta)\) of numbers of eigenvalues with positive, negative and zero real part.

Taussky wrote on generalizations of Lyapunov’s theorem in the early 1960s, and her survey articles brought the subject to the attention of matrix theorists. It is now central in control theory.

Commuting matrices and property L

For the general reader. Two matrices need not commute. When they do, they share eigenvectors, and the eigenvalues of any combination of them are the same combination of their eigenvalues. Taussky asked how much of this survives without commutativity. With Theodore Motzkin she studied pairs whose combinations still have “linear” eigenvalues, and showed that under mild conditions such pairs must in fact commute.

Proof sketch. \(B\) maps each eigenspace of \(A\) to itself, since \(A(Bv)=BAv=\lambda Bv\). So \(B\) has an eigenvector inside an eigenspace of \(A\): a common eigenvector \(v\). Extend \(v/|v|\) to an orthonormal basis and induct on the quotient \(\mathbb C^n/\mathbb Cv\).

A pair \((A,B)\) has property L if there are orderings \(\lambda_1,\dots,\lambda_n\) and \(\mu_1,\dots,\mu_n\) of their eigenvalues such that for all \(a,b\in\mathbb C\) the eigenvalues of \(aA+bB\) are \(a\lambda_i+b\mu_i\). Commuting pairs have property L. So do simultaneously triangularizable pairs, which need not commute.

Their papers also showed that property L is strictly weaker than simultaneous triangularizability for \(n\ge3\), and studied the algebraic variety of pairs with property L. The theme of what she called commutativity in finite matrices ran through much of her later work.

Integral matrices and ideal classes

For the general reader. Two integer matrices may have the same eigenvalues without being related by a change of basis that uses only integers. In 1933 Claiborne Latimer and Cyrus MacDuffee found that the different possibilities are counted by the class number of a ring of algebraic integers. Taussky gave the theorem its modern form and made it a central example of how matrices and number theory meet.

Let \(f\in\mathbb Z[x]\) be monic and irreducible of degree \(n\), \(\alpha\) a root, \(K=\mathbb Q(\alpha)\), and \(R=\mathbb Z[\alpha]\). Two matrices \(A,B\in M_n(\mathbb Z)\) are integrally similar if \(B=U^{-1}AU\) with \(U\in\mathrm{GL}_n(\mathbb Z)\). Two fractional \(R\)-ideals (non-zero finitely generated \(R\)-submodules of \(K\)) are equivalent if one is a non-zero multiple of the other by an element of \(K\).

The correspondence. Given \(A\), choose an eigenvector \(v\in R^n\) with \(Av=\alpha v\) (possible since \(\alpha\) is a simple eigenvalue and \(A\) has integer entries; clear denominators). Its entries \(v_1,\dots,v_n\) span a \(\mathbb Z\)-lattice \(I\subset K\) of rank \(n\), and \(\alpha I\subseteq I\) because \(\alpha v_i=\sum_ja_{ij}v_j\). So \(I\) is an \(R\)-ideal. Conversely, given an ideal with \(\mathbb Z\)-basis \(v_1,\dots,v_n\), multiplication by \(\alpha\) has an integer matrix \(A\) in this basis. Changing the basis by \(U\in\mathrm{GL}_n(\mathbb Z)\) conjugates \(A\) by \(U\), and scaling the ideal does not change \(A\).

Sums of squares

For the general reader. The product of two sums of two squares is again a sum of two squares; this is the rule \(|z|\,|w|=|zw|\) for complex numbers. The same holds for four squares (quaternions) and eight squares (octonions), and Adolf Hurwitz proved in 1898 that it fails for every other number of squares. Taussky’s 1970 survey Sums of squares, which won the Mathematical Association of America’s Lester R. Ford Award, traced this story and its connections to matrices, number theory and topology.

A composition formula of size \(n\) is an identity

where each \(z_k\) is a bilinear form in \(x\) and \(y\) with real coefficients.

Proof sketch. Write \(z=A(x)y\) with \(A(x)=x_1A_1+\cdots+x_nA_n\), \(A_i\in M_n(\mathbb R)\). The identity says \(A(x)^{\mathsf T}A(x)=|x|^2I\) for all \(x\), that is, \(A_i^{\mathsf T}A_i=I\) and \(A_i^{\mathsf T}A_j+A_j^{\mathsf T}A_i=0\) for \(i\neq j\). Replace \(A_i\) by \(B_i=A_n^{\mathsf T}A_i\), so that \(B_n=I\). Then for \(i<n\), \(B_i^{\mathsf T}=-B_i\), \(B_i^2=-I\), and \(B_iB_j=-B_jB_i\) for \(i\neq j<n\). So \(\mathbb R^n\) is a module over the Clifford algebra generated by \(n-1\) anticommuting square roots of \(-1\). Counting the dimensions of irreducible modules of these Clifford algebras gives \(n\in\{1,2,4,8\}\). The first case to fail, \(n=3\), is elementary (Exercise 10.2).

Washington and Caltech, 1947–1977

In 1947 Taussky and Todd went to the United States, at first for a year. They spent time at the Institute for Advanced Study, where John von Neumann was building his computer, and then joined the National Bureau of Standards, where she worked as a mathematics consultant in its National Applied Mathematics Laboratories. The Bureau was then at the centre of early American computing. In 1951 she organized the first Symposium on Numerical Algebra, which became the long-running Gatlinburg series of conferences on numerical linear algebra. She and Todd spent 1955 at the Courant Institute in New York.

In 1957 both moved to the California Institute of Technology. Todd was appointed professor; Taussky was appointed research associate, an untenured position, although she taught and supervised doctoral students. She received tenure in 1963 and became a full professor in 1971. Her Caltech students included Lorraine Foster, the first woman to receive a Caltech PhD in mathematics, as well as Charles Johnson, Robert Thompson and Edward Bender. Students remembered her exacting standards for mathematical writing and her generosity with problems.

She won the Lester R. Ford Award in 1971 for Sums of squares and was named a Los Angeles Times Woman of the Year in 1963. She served on the council of the American Mathematical Society and as its vice-president, and earlier on the council of the London Mathematical Society.

Last years and legacy, 1977–1995

Taussky retired in 1977 as professor emerita and kept up an enormous mathematical correspondence. Honours came mostly late: election to the Austrian Academy of Sciences (1975) and the Bavarian Academy of Sciences (1985), the Austrian Cross of Honour for Science and Art, First Class (1978), the renewal of her Vienna doctorate (1980), the Association for Women in Mathematics’ Noether Lecture (1981), an honorary doctorate from the University of Southern California (1988), and fellowship of the American Association for the Advancement of Science (1991). She died in Pasadena on 7 October 1995, aged 89. John Todd died in 2007.

Legacy

• Matrix theory. More than anyone she made linear algebra a research field in its own right, connected to number theory, group theory and numerical analysis. The International Linear Algebra Society established the Taussky–Todd Lecture in honour of both, and in 2021 the ILAS Taussky–Todd Prize.
• Eigenvalue localization. Gershgorin’s theorem, which she rescued from obscurity, is a standard tool in numerical analysis, and her irreducibility refinement is the basis of the theory of irreducibly diagonally dominant matrices.
• Number theory. Taussky’s conditions A and B and the Scholz–Taussky method remain basic in the computation of capitulation and class field towers. Her group theorem on 2-groups is a standard lemma.
• Mentorship. She supervised Hanna Neumann in wartime Oxford and a generation of matrix theorists at Caltech.

Connections

People linked to Olga Taussky-Todd across this collection. A name without a link has no page here yet. See Olga Taussky-Todd in the network of all 337 lives →

Teachers and advisors

• Emmy Noether
• Philipp Furtwängler 1930

Students

• Charles R. Johnson
• Edward Bender
• Hanna Neumann 1942-1943
• Lorraine Foster
• Robert C. Thompson

Influenced by

• David Hilbert
• Emil Artin

Colleagues and collaborators

• Emmy Noether 1931-1935
• G. H. Hardy
• John von Neumann 1947
• Kurt Gödel
• Otto Hahn 1925-1934
• Richard Courant 1931-1932
• Arnold Scholz 1932-1934
• Karl Menger 1932-1934
• Theodore Motzkin 1952-1955
• Wilhelm Magnus 1931-1932

Family

• John Todd

Also mentioned in this life

• Richard Dedekind
• Georg Frobenius
• Évariste Galois
• Carl Friedrich Gauss
• Adolf Hurwitz
• Felix Klein
• Aleksandr Lyapunov
• Issai Schur

Fields

• algebraic number theory 9 lives
• group theory 9 lives
• numerical analysis 6 lives

On CoursesHub

• Group Theory · Physical Mathematics
• Numerical Methods & Computational Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Olga Taussky-Todd, MacTutor History of Mathematics.
2. Olga Taussky-Todd, Wikipedia.
3. Taussky-Todd, Olga, Neue Deutsche Biographie.
4. F. L. Bauer, Olga Taussky-Todd, obituary, Bavarian Academy of Sciences.
5. D. Madden, John Todd and Olga Taussky-Todd, Irish Math. Soc. Bulletin 69 (2012).

Mathematics

1. O. Taussky, Über eine Verschärfung des Hauptidealsatzes, J. reine angew. Math. 168 (1932).
2. A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. reine angew. Math. 171 (1934).
3. O. Taussky, A remark on the class field tower, J. London Math. Soc. 12 (1937).
4. O. Taussky, A recurring theorem on determinants, Amer. Math. Monthly 56 (1949).
5. O. Taussky, On a theorem of Latimer and MacDuffee, Canad. J. Math. 1 (1949).
6. T. S. Motzkin and O. Taussky, Pairs of matrices with property L, Trans. Amer. Math. Soc. 73 (1952) and 80 (1955).
7. O. Taussky, Matrices of rational integers, Bull. Amer. Math. Soc. 66 (1960).
8. O. Taussky, Sums of squares, Amer. Math. Monthly 77 (1970).
9. O. Taussky, A remark concerning Hilbert’s Theorem 94, J. reine angew. Math. 239/240 (1970).
10. S. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk SSSR (1931).
11. C. G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. of Math. 34 (1933).
12. R. S. Varga, Geršgorin and His Circles, Springer, 2004.
13. R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge, 2013.
14. A. R. Rajwade, Squares, Cambridge, 1993.
15. D. C. Mayer, p-Capitulation over number fields with p-class rank two, arXiv:1605.03695.